Problem: The product of three different positive integers is equal to $7^3$. What is the sum of the three integers?
All three integers must be divisors of $7^3$.  The only divisors of $7^3$ are $7^0$, $7^1$, $7^2$, $7^3$.  We have $7^3=7^0\times7^1\times7^2$. Thus our answer is $7^0+7^1+7^2=1+7+49=\boxed{57}$.